Inversion of the fundamental thermodynamic equations for isentropic nozzle flow analysis

Joseph Majdalani, Brian A. Maicke

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The isentropic flow equations relating the thermodynamic pressures, temperatures, and densities to their stagnation properties are solved in terms of the area expansion and specific heat ratios. These fundamental thermofluid relations are inverted asymptotically and presented to arbitrary order. Both subsonic and supersonic branches of the possible solutions are systematically identified and exacted. Furthermore, for each branch of solutions, two types of recursive approximations are provided: a property-specific formulation and a more general, universal representation that encompasses all three properties under consideration. In the case of the subsonic branch, the asymptotic series expansion is shown to be recoverable from Bürmann's theorem of classical analysis. Bosley's technique is then applied to verify the theoretical truncation order in each approximation. The final expressions enable us to estimate the pressure, temperature, and density for arbitrary area expansion and specific heat ratios with no intermediate Mach number calculation or iteration. The analytical framework is described in sufficient detail to facilitate its portability to other nonlinear and highly transcendental relations where closed-form solutions may be desirable.

Original languageEnglish (US)
Article number031201
JournalJournal of Engineering for Gas Turbines and Power
Volume134
Issue number3
DOIs
StatePublished - Jan 9 2012

Fingerprint

Nozzles
Thermodynamics
Specific heat
Mach number
Temperature

All Science Journal Classification (ASJC) codes

  • Nuclear Energy and Engineering
  • Fuel Technology
  • Aerospace Engineering
  • Energy Engineering and Power Technology
  • Mechanical Engineering

Cite this

@article{856abd2585724bfaab5d764d8de95abb,
title = "Inversion of the fundamental thermodynamic equations for isentropic nozzle flow analysis",
abstract = "The isentropic flow equations relating the thermodynamic pressures, temperatures, and densities to their stagnation properties are solved in terms of the area expansion and specific heat ratios. These fundamental thermofluid relations are inverted asymptotically and presented to arbitrary order. Both subsonic and supersonic branches of the possible solutions are systematically identified and exacted. Furthermore, for each branch of solutions, two types of recursive approximations are provided: a property-specific formulation and a more general, universal representation that encompasses all three properties under consideration. In the case of the subsonic branch, the asymptotic series expansion is shown to be recoverable from B{\"u}rmann's theorem of classical analysis. Bosley's technique is then applied to verify the theoretical truncation order in each approximation. The final expressions enable us to estimate the pressure, temperature, and density for arbitrary area expansion and specific heat ratios with no intermediate Mach number calculation or iteration. The analytical framework is described in sufficient detail to facilitate its portability to other nonlinear and highly transcendental relations where closed-form solutions may be desirable.",
author = "Joseph Majdalani and Maicke, {Brian A.}",
year = "2012",
month = "1",
day = "9",
doi = "10.1115/1.4003963",
language = "English (US)",
volume = "134",
journal = "Journal of Engineering for Gas Turbines and Power",
issn = "0742-4795",
publisher = "American Society of Mechanical Engineers(ASME)",
number = "3",

}

Inversion of the fundamental thermodynamic equations for isentropic nozzle flow analysis. / Majdalani, Joseph; Maicke, Brian A.

In: Journal of Engineering for Gas Turbines and Power, Vol. 134, No. 3, 031201, 09.01.2012.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Inversion of the fundamental thermodynamic equations for isentropic nozzle flow analysis

AU - Majdalani, Joseph

AU - Maicke, Brian A.

PY - 2012/1/9

Y1 - 2012/1/9

N2 - The isentropic flow equations relating the thermodynamic pressures, temperatures, and densities to their stagnation properties are solved in terms of the area expansion and specific heat ratios. These fundamental thermofluid relations are inverted asymptotically and presented to arbitrary order. Both subsonic and supersonic branches of the possible solutions are systematically identified and exacted. Furthermore, for each branch of solutions, two types of recursive approximations are provided: a property-specific formulation and a more general, universal representation that encompasses all three properties under consideration. In the case of the subsonic branch, the asymptotic series expansion is shown to be recoverable from Bürmann's theorem of classical analysis. Bosley's technique is then applied to verify the theoretical truncation order in each approximation. The final expressions enable us to estimate the pressure, temperature, and density for arbitrary area expansion and specific heat ratios with no intermediate Mach number calculation or iteration. The analytical framework is described in sufficient detail to facilitate its portability to other nonlinear and highly transcendental relations where closed-form solutions may be desirable.

AB - The isentropic flow equations relating the thermodynamic pressures, temperatures, and densities to their stagnation properties are solved in terms of the area expansion and specific heat ratios. These fundamental thermofluid relations are inverted asymptotically and presented to arbitrary order. Both subsonic and supersonic branches of the possible solutions are systematically identified and exacted. Furthermore, for each branch of solutions, two types of recursive approximations are provided: a property-specific formulation and a more general, universal representation that encompasses all three properties under consideration. In the case of the subsonic branch, the asymptotic series expansion is shown to be recoverable from Bürmann's theorem of classical analysis. Bosley's technique is then applied to verify the theoretical truncation order in each approximation. The final expressions enable us to estimate the pressure, temperature, and density for arbitrary area expansion and specific heat ratios with no intermediate Mach number calculation or iteration. The analytical framework is described in sufficient detail to facilitate its portability to other nonlinear and highly transcendental relations where closed-form solutions may be desirable.

UR - http://www.scopus.com/inward/record.url?scp=84855381183&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855381183&partnerID=8YFLogxK

U2 - 10.1115/1.4003963

DO - 10.1115/1.4003963

M3 - Article

AN - SCOPUS:84855381183

VL - 134

JO - Journal of Engineering for Gas Turbines and Power

JF - Journal of Engineering for Gas Turbines and Power

SN - 0742-4795

IS - 3

M1 - 031201

ER -